'supper' is not as common a term for the evening meal in AmE (my family used to call it that when I was a kid but I have rarely heard anybody else use it).

'dinner' or 'supper' is the evening meal 5-7pm, or if later than that, more likely to be called a 'late dinner' or 'late supper'. That is, in AmE, 'lunch' = midday meal a midday meal is never called dinner or supper (but see the exception below). If you eat your midday meal at 3pm, that's kind of a late lunch, but it wouldn't be called anything else. 'lunch' is the midday meal (11:30am-1:30pm), however large it is (if you're eating something around that time, and you don't eat something bigger around that time, that was your lunch). Much Later Edit: I happened across this article discussing the agricultural roots of midday dinner and evening supper, just to add a bit more to the conversation.

So whether you use lunch/dinner or dinner/supper is heavily determined by when your culture traditionally has its largest meal. Lunch is almost the midday equivalent of supper - it's also a lighter and less formal meal than Dinner, but is used specifically when referring to a midday meal. Rooted in the word 'to sup', it comes, again, from farming traditions - many farming families would have a pot of soup cooking throughout the day, and would eat it in the evening - specifically, they would 'sup' the soup. Supper is more specifically a lighter evening meal. For instance, many people who grew up in the American South and/or on farms traditionally ate larger meals at noontime to give them the strength to keep working through the afternoon.

Whether it takes place at noon or in the evening is mostly a cultural thing. Dinner is considered to be the 'main' or largest meal of the day.

What time is the afternoon